Let $\mathcal{A}$ be a standard operator algebra on a Banach space $\mathcal{X}$ with $\dim \mathcal{X}\geq 2$. In this paper, we characterize the forms of additive maps on $\mathcal{A}$ which strongly preserve the square zero of $ \lambda $-Lie product of operators, i.e., if $\phi:\mathcal{A}\longrightarrow \mathcal{A}$ is an additive map which satisfies $$ [A,B]^2_{\lambda}=0 \Rightarrow [\phi(A),B]^2_{\lambda}=0,$$ for every $A,B \in \mathcal{A}$ and for a scalar number $\lambda$ with $\lambda \neq -1$, then it is shown that there exists a function $\sigma: \mathcal{A} \rightarrow \mathbb{C}$ such that $\phi(A)= \sigma(A) A$, for every $A \in \mathcal{A}$.
